theorem
  for P,A,B st 0 < P.B holds (P.|.B.A = P.A iff A,B
  are_independent_respect_to P)
proof
  let P,A,B;
  assume
A1: 0 < P.B;
  thus P.|.B.A = P.A implies A,B are_independent_respect_to P
  proof
    assume P.|.B.A = P.A;
    then P.(A /\ B)/P.B * P.B = P.A * P.B by A1,Def6;
    then P.(A /\ B) = P.A * P.B by A1,XCMPLX_1:87;
    hence thesis;
  end;
  assume A,B are_independent_respect_to P;
  then P.(A /\ B) * (P.B)" = P.A * P.B * (P.B)";
  then P.(A /\ B) * (P.B)" = P.A * (P.B * (P.B)");
  then P.(A /\ B) * (P.B)" = P.A * 1 by A1,XCMPLX_0:def 7;
  then P.(A /\ B)/P.B = P.A by XCMPLX_0:def 9;
  hence thesis by A1,Def6;
end;
